Abstract

The factorization rules of Li [J. Opt. Soc. Am. A 13, 1870 (1996)] are generalized to a cylindrical geometry requiring the use of a Bessel function basis. A theoretical study confirms the validity of the Laurent rule when a product of two continuous functions or of one continuous and one discontinuous function is factorized. The necessity of applying the so-called inverse rule in factorizing a continuous product of two discontinuous functions in a truncated basis is demonstrated theoretically and numerically.

© 2004 Optical Society of America

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References

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  1. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  2. J. Jarem, “Rigorous coupled wave analysis of radially and azimuthally-inhomogeneous, elliptical, cylindrical systems,” Prog. Electromagn. Res. (PIER) 34, 89–115 (2001).
    [CrossRef]
  3. P. Banerjee, J. Jarem, “Convergence of electromagnetic field components across discontinuous permittivity profiles,” J. Opt. Soc. Am. A 17, 491–492 (2000).
    [CrossRef]
  4. P. Lalanne, G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  5. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  6. L. Schwartz, Méthodes Mathématiques pour les Sciences Physiques (Hermann, Paris, 1961), p. 174.
  7. M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

2001 (1)

J. Jarem, “Rigorous coupled wave analysis of radially and azimuthally-inhomogeneous, elliptical, cylindrical systems,” Prog. Electromagn. Res. (PIER) 34, 89–115 (2001).
[CrossRef]

2000 (1)

1996 (3)

Banerjee, P.

Granet, G.

Guizal, B.

Jarem, J.

J. Jarem, “Rigorous coupled wave analysis of radially and azimuthally-inhomogeneous, elliptical, cylindrical systems,” Prog. Electromagn. Res. (PIER) 34, 89–115 (2001).
[CrossRef]

P. Banerjee, J. Jarem, “Convergence of electromagnetic field components across discontinuous permittivity profiles,” J. Opt. Soc. Am. A 17, 491–492 (2000).
[CrossRef]

Lalanne, P.

Li, L.

Morris, G.

Nevière, M.

M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

Popov, E.

M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

Schwartz, L.

L. Schwartz, Méthodes Mathématiques pour les Sciences Physiques (Hermann, Paris, 1961), p. 174.

J. Opt. Soc. Am. A (4)

Prog. Electromagn. Res. (PIER) (1)

J. Jarem, “Rigorous coupled wave analysis of radially and azimuthally-inhomogeneous, elliptical, cylindrical systems,” Prog. Electromagn. Res. (PIER) 34, 89–115 (2001).
[CrossRef]

Other (2)

L. Schwartz, Méthodes Mathématiques pour les Sciences Physiques (Hermann, Paris, 1961), p. 174.

M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

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Figures (3)

Fig. 1
Fig. 1

Example 1. Convergence with respect to kMax=0.1 M of the coefficient hn,n of the function hn(r)=Jn(r), calculated with the direct and the inverse rules for several values of n, shown in the insert.

Fig. 2
Fig. 2

Example 2. Convergence with respect to kMax=0.1 M of the coefficients (a) h1,1 and (b) h1,5 of the function hn(r)=1/r.

Fig. 3
Fig. 3

Example 3. (a) Reconstruction of h0(r) with the direct rule with three different values of kMax, together with the exact plot, Eq. (46). (b) The same as in (a) but with kMax=200. Gibbs phenomena are still observed. (c) The same as in (a) but with the inverse rule. The exact values cannot be distinguished from the curve obtained with kMax=20. Gibbs phenomena are completely absent.

Equations (60)

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h(x)=g(x)f(x).
f(x)=m=1fmφm(x).
f(M)(x)=m=1Mfmφm(x).
fm=f(x)|φm(x),
φn(x)|φm(x)=δnm.
af(x)+bg(x)|φm(x)=af(x)|φm(x)+bg(x)|φm(x).
f(x+d)=f(x)exp(iα0d),
φm(x)=exp(iαmx)expiα0+m 2πdx,
f|φm=1d 0df(x)φ¯m(x)dx,
hm=pgmpfp,
gmp=g(x)φp(x)|φm(x).
gmp=1d 0dg(x)expi(p-m) 2πdx.
hm=g(x)f(x)|φm(x)=g(x)p=1fpφp(x)φm(x)=p=1fpg(x)φp(x)|φm(x)=p=1gmpfp.
hm(M)=p=1Mgmpfp,m,p=1:M,
hm(M)=p=1M(ginv(M))mp-1fp,
ginv,mp=1g(x)φp(x)φm(x),m,p=1:M.
hm(M)=g(x)f(M)(x)|φm(x)=g(x)p=1Mfpφp(x)φm(x)=p=1Mfpg(x)φp(x)|φm(x)=p=1Mgmpfp,1mM.
fm(M)=h(x)/g(x)|φm(x)=1g(x) p=1Mhpφp(x)φm(x)=p=1Mhp1g(x)φp(x)φm(x)=p=1Mginv,mphp.
f(r, θ, z)=n=-+fn(r, z)exp(inθ).
fn(r, z)=0kdkf˜n(k, z)Jn(kr).
f(r, θ, z)=0kdkn=-+f˜n(k, z)Jn(kr)exp(inθ).
fˆn(k, z)=kf˜n(k, z),
f(r, θ, z)=0dkn=-+fˆn(k, z)Jn(kr)exp(inθ).
0rdrJn(kr)Jn(kr)=δ(k-k)k,
f˜n(k)=0rdrfn(r)Jn(kr),
fˆn(k)=kf˜n(k)=k0rdrfn(r)Jn(kr).
fn(r)|Jn(kr)=k0rdrfn(r)Jn(kr).
fn(r)=0kdkf˜n(k, z)Jn(kr)
fn(r)=m=1kmΔkf˜n,mJn(kmr)=m=1fn,mJn(kmr),
f˜n,m=0rdrfn(r)Jn(kmr),
fn,m=Δkkmf˜n,m.
gn,mp=Δkkm0rgn(r)Jn(kpr)Jn(kmr)dr.
g(r)=1,r>RQ,r<R,
hn(r)=Jn(kpr).
hn,mΔkkm0hn(r)Jn(kmr)rdr=km δ(km-kp)kmΔkΔk0δmp.
δ(km-kp)F(km)dkm=F(kp)discretizationmδmpFm=Fp,F(kp)Fp.
gn,mqΔkkm0gn(r)Jn(kqr)Jn(kmr)rdr=Δkkm0Jn(kqr)Jn(kmr)rdr+(Q-1)Δkkm0RJn(kqr)Jn(kmr)rdr=δmq+(Q-1)In,mq,
In,mq=Δkkm0RJn(kqr)Jn(kmr)rdr
 ginv,n,mqΔkkm0 1gn(r)Jn(kqr)Jn(kmr)rdr=Δkkm0Jn(kqr)Jn(kmr)rdr+1Q-1Δkkm0RJn(kqr)Jn(kmr)rdr=δmq+1Q-1In,mq.
fn,m=1Q 0RJn(kmr)Jn(kpr)rdrΔkkm+RJn(kmr)Jn(kpr)rdrΔkkm=δmp+1Q-1In,mpginv,n,mp.
hn,m(M,direct)=q=1Mgn,mq(M)fn,q(M)=δmp+(Q-1)1-1Q×In,mp-q=1MIn,mqIn,qp,
r0kdkJn(kr1)Jn(kr2)=δ(r1-r2),
q=1MIn,mqIn,qpΔkkm0kqdkq0Rr1dr1Jn(kmr1)Jn(kqr1)×0Rr2dr2Jn(kqr2)Jn(kpr2)=Δkkm0Rr1dr1Jn(kmr1)0Rr2dr2Jn(kpr2)×δ(r1-r2)r2=Δkkm0Rr1dr1Jn(kmr1)Jn(kpr1)=In,mp.
hn,m(M,direct)δmp=hn,m.
hn,m(M,inv)=q=1M[ginv,n(M)]mq-1fn,q(M)=Eq.(35)q=1M[ginv,n(M)]mq-1ginv,n,qp(M)=δmphn,m,
hn(r)=1r,
fn(r)=1gn(r)hn(r).
hn,mΔkkm0 1rJn(kmr)rdr=Δkkm 1km=Δk.
fn(r)Jn(kr(1)r),r<R,
fn(r)Jn(kr(2)r),r>R,
kr(1)2=k02ν(1)2-kz2,
kr(2)2=k02ν(2)2-kz2,
fn(r)=Jn(kpr), r>R,
fn(r)=1Q Jn(kpR)Jn(ksR)Jn(ksr),r<R,
hn(r)=Jn(kpr),r>R,
hn(r)=Jn(kpR)Jn(ksR)Jn(ksr),r<R.
gn,mq=δmq+(Q-1)In,mq,
fn,m=δmp-1QIn,mp+Jn(kpR)Jn(ksR)In,ms,
hn,m=δmp-In,mp+Jn(kpR)Jn(ksR)In,ms.
hn(M)(r)=m=1Mhn,m(M)Jn(kmr).

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